 -prolongation 40
 -prolongation of a vector field 38
0-contact form 35
Action function 42
Adapted chart to a distribution 24
Alternative Lagrangians 123
Anderson — Duchamp — Krupka conditions 61
Angular momentum, conservation law 159 165
Anholonomic prolongation of a fibered manifold 30
Annihilator 21
Aristotle group 159
Associated fiber chart 30
Associated fibered coordinates 30 31
Associated system of a closed two-form 25
Autonomous Lagrangean system 71 210
Autonomous semi-finslerian structure 223
Base of a fibered manifold 29
Brachystochrone 3 4
Brachystochrone problem 3
Bunch of final constraint submanifolds 133
Bundle of  -vertical vectors 31
Canonical connection 222
Canonical dynamical two-form 221
Canonical form of a Lagrangean system 67
Canonical form of a Lepagean 2-form 67
Canonical form of Hamilton equations 81 107 108 145
Canonical transformation 187
Cartan connection 224
Cartan distribution 34 37 39
Cartan form 8 47 48 209
Cartan symmetry 172
Cartan theorem 26
Cauchy characteristic 25 191
Cauchy characteristic, vector field 25 191
Characteristic distribution of a closed two-form 25 175
Characteristic distribution of a Lagrangean system 88 107 108 146
Christoffel symbols 121 224
Classical Hamilton — Jacobi equation 18 174 189
Classical Hamilton — Jacobi integration method 18 197
Classification of Lagrangean systems 130
Classification of symmetries 150
Classification of variational semispray connections 227
Codistribution 22
Compatibility conditions 38
Complementary distributions 21
Complete integral 18 174 175
Complete integral of a closed 2-form 176
Complete set of independent first integrals 23 26
Completely integrable connection 38
Completely integrable distribution 23 207
Completely integrable generalized connection 40
Components of a jet field 38
Configuration space 31 63 151
Connection on a fibered manifold 37 38 183 195
Conservation law 149 152 157
Constant of the motion 149 152 160
Constrained system 130
Constraint algorithm 25 130 131 132 134 172 210
Contact form 34
Contact transformation 212
Contactization 34
Contactness 32
Continuous distribution 21
Coordinate free Liouville Theorem 179
Coriolis force 14
Cosymplectic Lagrangean system 99 210
Cotangent representation of symplectic Lagrangean systems 218
Critical section of a Lagrangian 50 51
Curvature of a connection 38
Darboux chart 26 175
Darboux theorem 26 175 176
Decomposition theorem 35
Deformation of a section 42 49
Degenerate Lagrangean system 129
Degree of freedom 12 78
Dirac theory of constrained systems 103 113 134 218 219
distribution 21
Distribution of a constant rank 21 22
Distribution of a locally constant rank 22
Distribution of a non-constrant rank 22
Distribution of class  1
Dynamical form 37 44 53 116
Dynamical symmetry 151 171
Eikonal equation 191
Embedding theorem 191
Energy, conservation law 159 165
Energy, one-form 211
Equivalent dynamical forms 117 126
Equivalent Lagrangians 10 61 149
Equivalent systems of a Hamiltonian and momenta 70
Euclidean group 159 162 162 165 168 169
Euler — Lagrange distribution 90 107 108 146 147
Euler — Lagrange distribution, constrained to a submanifold 133
Euler — Lagrange equations 1 12 41 49 51 53 80 209
Euler — Lagrange expressions 3 46
Euler — Lagrange form 41 46 49 209
Euler — Lagrange mapping 41 46 64
Even-order Lagrangean system 100 105
Even-order variational equations 105 183 195
Evolution of a function 100 107 109 214
Exact integration methods for variational equations 148 149 174
Extended dynamics 81 87 129 132 148 217
Extended Lagrangian 85 88 108 109
Extremal of a Lagrangian 50 51 53
Extremal of a locally variational form 53
Fiber chart 29
Fibered manifold 29
Field of extremals 174 190 192
Field of Lagrangean submanifolds 181
Final constraint submanifold 133 218
Finite symmetry 24
Finsler manifold 220 220 223
Finslerian structure 223
First integral 18 23 100 149 152
First integral of a Lagrangean system 152
First integrals in involution 175 177 178
First Theorem of Emmy Noether 157
First variation 2 41
First variation formula 2 41 42 43 47
First variation of the action function 42
First-order Lagrangean system 74 75 101 102 118 122 145 192 214 220
Flat connection 38
Flat generalized connection 40
Foliation 23
Force on a fibered manifold 225
Free energy 12
Free particle on a semi-finslerian manifold 225
Frobenius theorem 24
Froelicher — Nijenhuis bracket 38
Fundamental Lepagean equivalent 49
Galilei group 159 162 166 167 168 169
Galilei transformation 159 163
Gauge function 61 68
General Liouville — Jacobi Theorem 189
Generalized connection 37 40 141 181 181 183
Generalized coordinates 12 78
Generalized dynamical symmetry 172
Generalized Hamilton equations 84 86 88 91 130 145 191 210
Generalized Hamilton — Jacobi equation 180 189
Generalized horizontal form 40
Generalized jet field 40
Generalized Legendre chart 145
Generalized Legendre coordinates 145 214
Generalized Legendre transformation 130 145 148 214
Generalized Liouville Theorem 178 179
Generalized Noether Theorem 160 160 172
Generalized Poincre — Cartan form 49
Generalized Van Hove Theorem 191
Generating function of a canonical transformation 174 187 188
Generating function of acontact transformation 212
| Generators of a distribution 21
Geodesic distance in a field of extremals 200
Geodesic distance in a Hamilton — Jacobi field 200
Geodesies 10
Geometric classification of Lagrangean systems 96 129 130
Global extended dynamics 131
Global inverse problem of the calculus of variations 64 65
Global Lagrangian 54 65
Global proper dynamics 131
Globally variational form 53 54
Grifone connection 223
Hamilton equations 17 80 84 107 108 145 210 211 213
Hamilton equations for autonomous Lagrangean systems 211
Hamilton equations for semiregular autonomous Lagrangean systems 212
Hamilton form 84 107 108
Hamilton function 17 67 70 71 210 210
Hamilton vector field 90 191 214
Hamilton — Jacobi distribution 192 193 194 195 196
Hamilton — Jacobi equation 18 187
Hamilton — Jacobi equation for fields of extremals 190
Hamilton — Jacobi field 193 194 195
Hamilton — Jacobi integration method 18 174 182 197
Hamilton's variational principle 12 13 15
Hamilton, extremal 84 129 194 212
Hamiltonian 17 67 70 71 210 210
Hamiltonian formalism 82 217
Hamiltonian system 84 88 100 106
Helmholtz conditions 5 8 57 61 64 118 161
Higher-order semispray problem 96 129 132 134 141 212
Hilbert — Einstein Lagrangian 113
Holonomic constraints 12 77
Holonomic Hamilton extremal 86 87 93 98
Holonomic section 30 34
Homomorphism of fibered manifolds 30
Horizontal distribution 32 37 38
Horizontal form 32
Horizontal form of a connection 38
Horizontality 32
Horizontalization 33
Image of the Euler — Lagrange mapping 64
Independent first integrals 23 175
Infinitesimal first variation formula 47
Infinitesimal symmetry 24
Integral first variation formula 47
Integral manifold 23
Integral manifold of maximal dimension 23
Integral mapping of a distribution 22
Integral mapping of a distribution of maximal dimension 23
Integral section of a connection 38
Integral section of a dynamical form 53 116
Integral section of a jet field 37
Integral section of an ideal 189
Integrating factor 118 126
Integration by quadratures 18 175
Integration methods 16 142 148 188 195 207
Integration problem 18 95 129 132 149 157 174
Invariant transformation of a differential form 26
Invariant transformation of a distribution 24
Inverse Legendre transformation 106
Inverse problem of the calculus of variations 4 9 16 61 65 118 123 126 150 180 207
Inverse problem of the calculus of variations for distributions 123
Isomorphism of a fibered manifold 30 75
Isoperimetric problem 1
Jacobi chart 183 196
Jacobi complete integral 183
Jacobi theorem 180 182 183
Jacobi Theorem for even-order variational equations 183
Jacobi Theorem for odd-order variational equations 184
Jacobi Theorem for regular even-order variational equations 186
Jet field 37
Jet manifold 29
Jet prolongation of a fibered manifold 29
k-contact form 35
Kernel of the Euler-Lagrange mapping 64
kinetic energy 12 122 163
Kinetic energy of a metric 221
Lagrange function 1 41 210
Lagrange space 220 223
Lagrangean foliation 181
Lagrangean submanifold 181 196
Lagrangean system 63 81 83 97 126 149
Lagrangean system of order zero 73 75 83 88 90 100 101 102 109 127
Lagrangean system with dynamical constraints 212
Lagrangean system with no dynamical constraints 153
Lagrangean system with no primary dynamical constraints 131
Lagrangean system with no primary semispray constraints 132
Lagrangean system with no secondary semispray constraints 133
Lagrangean system with primary dynamical constraints 131
Lagrangean system with primary semispray constraints 131
Lagrangean system with secondary semispray constraints 133
Lagrangean system, admitting generalized Legendre transformations 144 214 218
Lagrangian 37 41 210
Lagrangian affine in accelerations 74 102
Lagrangian affine in velocities 73 100 101 109
Lagrangian on  2\0
Legendre chart 105
Legendre coordinates 17 105 148 213
Legendre transformation 17 97 103 105 144 148 213
Lepage mapping of the first kind 46
Lepage mapping of the second kind 58
Lepagean equivalent of a Lagrangian 46
Lepagean equivalent of a locally variational form 58
Lepagean n-form 41 48
Lepagean one-form 41 46 56
Lepagean two-form 55 55 55 81 83 97 150 171 208 209
Levi-Civita connection 224
Line element induced by a minimal-order Lagrangian 200
Liouville form 211
Liouville integration method 18 207
Liouville theorem 175 178 179 182
Local inverse problem of the calculus of variations 61 98 118
Local Lagrangian 54 55
Local Lagrangian section of a fibered manifold 29
Locally variational form 53 54 81 118 161 220
Lorentz force 14 122 226
Manifold of s-jets of local sections 29
Manifold of s-velocities 31
Maximal integral manifold 23
Maximal vertical distribution 32
mc-parameter family of Hamilton-Jacobi distributions 197
Mechanical system 117 126 225
Mechanical system on a semi-finslerian manifold 225
Method of canonical transformations 207
Metric on  220
Minimal-order Lagrangian 68 158
Minkowski metric 208
Minkowski space-time 170
Momentum 17 67 70 144
Momentum, conservation law 159 165
Momentum, map 150
Multiplier 118
Newton equations 54 80 121
Newtonian determinism 99
Noether equation 155 161
Noether symmetry 172
Noether theorem 157 158 172
Noether — Bessel — Hagen equation 155 161
Noetherian symmetry 172
Non-holonomic constraints 78
Nonlinear connection 37
Odd-order differential equations 53
Odd-order Euler — Lagrange equations 68 100 102 103 105 184
Odd-order Lagrangean system 68 103 105 183 186 195 197 198 198 217
Odd-order variational equations 68 100 102 103 105 184
of a vector field 31
One-contact form 35
One-parameter family of sections 49
Order of a Lagrangean system 63 63
Order of contactness 36
Order-reduction problem for Lagrangians 56 64
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